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Lect25_OrbitalAngMom - The General Theory of Angular...

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Lecture 25: Orbital Angular Momentum PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore The General Theory of Angular Momentum Starting point: Assume you have three operators that satisfy the commutation relations: Let: Conclusions: Simultaneous eigenstates of J 2 and J z exist They must satisfy: Where the quantum numbers take on the values: y x z y x iJ J J J J J J ± = + + = ± 2 2 2 2 y x z x z y z y x J i J J J i J J J i J J h h h = = = ] , [ ] , [ ] , [ m j m m j J m j j j m j J z , , , ) 1 ( , 2 2 h h = + = 1 , ) 1 ( ) 1 ( , ± ± ! + = ± m j m m j j m j J h j j j j j m j , 1 , , 2 , 1 , , 3 , 2 5 , 2 , 2 3 , 1 , 2 1 , 0 ! + ! + ! ! = = K K
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Orbital Angular Momentum For orbital angular momentum we have: So that: In coordinate representation we have: P R L r r v ! = x y z YP XP L ! = v L " # i h r r $ r % ( ) ! " " ! " # # # # # # = $ % sin 1 1 0 0 det r r r r e e e r r r r r r r ! " ! " ! # # + # # $ = e e r r sin 1 Continued Note that it doesn’t depend on r or ! / ! r . ! ! " # $ $ % & + ( ( = ) * ) * ) e e i L r r h r sin 1 ! ! " # $ $ % & + ( ) ! ! " # $ $ % & + ( ( = ) = * + * * + * + * + * e e e e L L L r r r r h r r sin 1 sin 1 2 2 ! " ! " ! " !
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